Expected number and distribution of critical points of real Lefschetz pencils
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1085-1113.

We give an asymptotic probabilistic real Riemann–Hurwitz formula computing the expected real ramification index of a random covering over the Riemann sphere. More generally, we study the asymptotic expected number and distribution of critical points of a random real Lefschetz pencil over a smooth real algebraic variety. Throughout the paper, we give similar results for the complex case. Our main tool is Hörmander theory of peak sections.

Dans cet article, on donne une formule de Riemann–Hurwitz asymptotique et probabiliste qui calcule la valeur attendue de l’indice de ramification réel d’un revêtement aléatoire de la sphère de Riemann. Plus généralement, on étudie l’asymptotique de la valeur attendue du nombre et de la distribution des points critiques réels d’un pinceau de Lefschetz réel sur une variété algébrique réelle. Tout au long de l’article, on donne des résultats analogues pour le cas complexe. Notre outil principal est la théorie des sections pics d’Hörmander.

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DOI: 10.5802/aif.3331
Classification: 14P99,  32U40,  60D05
Keywords: real algebraic varieties, Lefschetz pencils, peak sections, random geometry
Ancona, Michele 1

1 Tel Aviv University School of Mathematical Sciences Tel Aviv-Yafo (Israël)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ancona, Michele. Expected number and distribution of critical points of real Lefschetz pencils. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1085-1113. doi : 10.5802/aif.3331. https://aif.centre-mersenne.org/articles/10.5802/aif.3331/

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